{
  "id": "2009.03470",
  "version": "v1",
  "published": "2020-09-08T01:02:10.000Z",
  "updated": "2020-09-08T01:02:10.000Z",
  "title": "$l_p$ regularization for ensemble Kalman inversion",
  "authors": [
    "Yoonsang Lee"
  ],
  "comment": "22 pages, 7 figures",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.OC"
  ],
  "abstract": "Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to a compact set, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in the standard EKI. This paper proposes a strategy to implement $l_p, 0<p\\leq 1,$ regularization for EKI to recover sparse structures in the solution. The strategy transforms a $l_p$ problem into a $l_2$ problem, which is then solved by Tikhonov EKI. The transformation is explicit, and thus the proposed approach has a computational cost comparable to Tikhonov EKI. We validate the proposed approach's effectiveness and robustness through a suite of numerical experiments, including compressive sensing and subsurface flow inverse problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-08T01:02:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65J20",
      "65C05",
      "35Q93",
      "49M41"
    ],
    "keywords": [
      "ensemble kalman inversion",
      "regularization",
      "tikhonov eki",
      "subsurface flow inverse problems",
      "derivative-free optimization method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}